Invariants
| Level: | $56$ | $\SL_2$-level: | $8$ | Newform level: | $1568$ | ||
| Index: | $96$ | $\PSL_2$-index: | $48$ | ||||
| Genus: | $1 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (none of which are rational) | Cusp widths | $4^{4}\cdot8^{4}$ | Cusp orbits | $2^{2}\cdot4$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $1$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 8F1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 56.96.1.594 |
Level structure
| $\GL_2(\Z/56\Z)$-generators: | $\begin{bmatrix}3&10\\6&11\end{bmatrix}$, $\begin{bmatrix}22&39\\51&26\end{bmatrix}$, $\begin{bmatrix}51&4\\44&31\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 56.48.1.ep.1 for the level structure with $-I$) |
| Cyclic 56-isogeny field degree: | $16$ |
| Cyclic 56-torsion field degree: | $384$ |
| Full 56-torsion field degree: | $32256$ |
Jacobian
| Conductor: | $2^{5}\cdot7^{2}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 1568.2.a.e |
Rational points
This modular curve has infinitely many rational points, including 1 stored non-cuspidal point.
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 8.48.0-8.x.1.4 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 56.48.0-8.x.1.1 | $56$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 56.48.0-56.x.1.2 | $56$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 56.48.0-56.x.1.5 | $56$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 56.48.1-56.m.1.3 | $56$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 56.48.1-56.m.1.8 | $56$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 56.768.25-56.lv.1.2 | $56$ | $8$ | $8$ | $25$ | $10$ | $1^{20}\cdot2^{2}$ |
| 56.2016.73-56.bjn.1.4 | $56$ | $21$ | $21$ | $73$ | $40$ | $1^{16}\cdot2^{26}\cdot4$ |
| 56.2688.97-56.bit.1.5 | $56$ | $28$ | $28$ | $97$ | $49$ | $1^{36}\cdot2^{28}\cdot4$ |
| 112.192.3-112.ex.1.4 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 112.192.3-112.ez.1.4 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 112.192.3-112.he.1.4 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 112.192.3-112.hk.1.4 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 112.192.3-112.jc.1.4 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 112.192.3-112.ji.1.4 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 112.192.3-112.ka.1.4 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 112.192.3-112.kc.1.4 | $112$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.288.9-168.eaj.1.10 | $168$ | $3$ | $3$ | $9$ | $?$ | not computed |
| 168.384.9-168.bjy.1.6 | $168$ | $4$ | $4$ | $9$ | $?$ | not computed |
| 280.480.17-280.uv.1.4 | $280$ | $5$ | $5$ | $17$ | $?$ | not computed |